Page 195 - Foundations Of Differential Calculus

178 9. On Differential Equations
which we arrived at above in paragraph 288, the same relationship between
x and y is defined as that contained in the finite equation
2 2 x/a
y − ax = b e .

2
2 x/a
From this we have y = ax + b e , so that

2 x/a
ax + b e = y,
which is the function of x that y equals in the given differential equation.
2
2 x/a
Indeed, if we substitute the value ax + b e for y and if we substitute
its differential
b 2 x/a
adx + e dx
a
for 2ydy, we obtain the following identity:
2 2
b x/a b x/a
adx + e dx − adx − xdx − e dx + xdx =0.
a a
Hence it is clear that every differential equation exhibits the same relation-
ship between x and y as a certain finite relationship, which we can find
only with the aid of integral calculus.
298. In order that this may be understood more clearly, we suppose that
we know the function of x that is equal to y by reason of the differential
equation, whether of the first order or of higher order. We also let

dy = p dx, dp = q dx, dq = r dx,
2 2
etc., and if in the equation we take dx to be constant, then d y = qdx ,
3 3
d y = rdx , etc. When these values are substituted into the differential
equation, due to the homogeneous terms, there remains an equation con-
taining only finite quantities x, y, p, q, r, etc. Since p, q, r, etc. are quantities
that naturally depend on the function y, there really remains an equation
between two variables x and y. In turn it should be clear that a certain
relationship between the variables x and y is determined by every differen-
tial equation. For this reason, if in the solution of some problem we obtain
a differential equation between x and y, we can consider this to be equiv-
alent to a relationship between x and y, just as if we had come to a finite
equation.
299. In this way any differential equation can be reduced to finite form, so
that it contains nothing but finite quantities when the differentials, that is,
the infinitely small quantities, are removed. Since y is some certain function
of x,ifwelet dy = pdx, dp = qdx, dq = rdx, etc., when some differential
is taken to be constant, the second and higher differentials are expressed